Can’t position Gradient Texture on Object (eyeball) properly
Your issue is position of UV map.

Tutorial (and you) are using UV texture coordinates, but your UV map is not placed in center of "UV Editor". Since you move the front part of eye in a center it matches.



BTW instead of UV I would rather use Texture Coordinates 

type Generated (Loc -0.5 / -0.5 / -1)



or easier type Object (Loc 0 / 0 / -1)